Analysis of Variance: Effects of Interchanging Main Effects and Interactions by Rearranging Data

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Abstract
Fictitious data in a 2 × 2 factorial design are analyzed in two alternative ways, both consistent with standard procedure. The alternative analyses give opposite results: hypotheses whose tests prove significant in one analysis are nonsignificant in the other. This problem arises when cell frequencies are constant with respect to one factor and proportional with respect to the other. We may interchange interaction and main effect tests by rearranging the data, thereby altering the results of testing a particular hypothesis since interaction tests weight all cells equally while main effect tests weight each simple effect involved by its cell frequencies. Uncritical reliance on conventional analysis is questioned.

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